Majorana neutrino electromagnetic properties are studied through neutral current coherent neutrinonucleus scattering. We focus on the potential of the recently planned COHERENT experiment at the Spallation Neutron Source to probe muon-neutrino magnetic moments. The resulting sensitivities are determined on the basis of chi(2) analysis employing realistic nuclear structure calculations in the context of the quasiparticle random phase approximation. We find that they can improve existing limits by half an order of magnitude. In addition, we show that these facilities allow for standard model precision tests in the low energy regime, with a competitive determination of the weak mixing angle. Finally, they also offer the capability to probe other electromagnetic neutrino properties, such as the neutrino charge radius. We illustrate our results for various choices of experimental setup and target material.